Visually Representing the Landscape of Mathematical Structures

Katherine Gravel Hayden Jananthan Jeremy Kepner Department of Mathematics Department of Mathematics Lincoln Laboratory Supercomputing Center Massachusetts Institute of Technology Vanderbilt University Massachusetts Institute of Technology Cambridge, Massachusetts Nashville, Tennessee Lexington, Massachusetts [email protected] [email protected] [email protected]

Abstract—The information technology explosion has dramati- basic undergraduate courses to higher level subjects including cally increased the application of new mathematical ideas and has analysis, abstract algebra, and topology, visual intuition and led to an increasing use of mathematics across a wide range of tools for understanding still exist but become much sparser. fields that have been traditionally labeled “pure” or “theoretical”. There is a critical need for tools to make these concepts readily The lack of abundant, clear, visual explanations in abstract accessible to a broader community. This paper details the creation of visual representations of mathematical structures mathematics occurs naturally since abstract mathematics seeks commonly found in pure mathematics to enable both students to remove any dependency on the physical world and strives to and professionals to rapidly understand the relationships among make logical statements based only on generally stated rules and between mathematical structures. Ten broad mathematical [7]. Since the human brain is wired to process information structures were identified and used to make eleven maps, with visually, it is difficult to see how abstract math and visual 187 unique structures in total. The paper presents a method and implementation for categorizing mathematical structures and intuition may intersect [8], [9]. Therefore, there is a need to for drawing relationship between mathematical structures that create a representative visual that is both abstracted, to mirror provides insight for a wide audience. The most in depth map is the nature of pure mathematics, and specific, so individuals available online for public use [1]. may be able to associate objects represented in the map with Keywords—mathematics visualization, mathematics education, something tangible [10], [11]. pure mathematics Visual tools with the above mentioned ideas in mind have I.INTRODUCTION been created prior. For example, some individuals, generally It is often helpful to package information into visuals, which students, make small concept maps of different structures in are easily understood [2]. Visuals tools are used to enhance specific mathematical disciplines, or make concept maps of a understanding in media, education, professional circles, and group of related disciplines and exchange them on internet almost every other field. Many times, one may find it more forums [12], [13]. Although those maps may be useful to difficult to understand nonvisual media, especially when media the individuals who created them, they are of less utility to is attempting to convey complex or spatial information, such learning communities since they generally do not contain more as a direction manual for setting up a couch that does not than four words per node and do not explain the relationships come with pictures. Visuals representations are also of great between included concepts. There are also more rigorous use in education, specifically in mathematics, where they are representations such as a project by Wolfram Alpha, that able to ground abstract concepts in a more tangible medium: allows users to enter a mathematical structure into a search the physical world [3]–[6]. function that will return properties of the object, as well as a In mathematics as a whole, there are many visual intuitions local map of what it is related to [14]. Furthermore, in the book arXiv:1809.05930v1 [math.HO] 16 Sep 2018 and tools, although they are concentrated around less abstract Mathematics of Big Data there is extensive use of depictions mathematics. In grade school, one is taught to count on one’s of the abstract mathematics necessary to define an associative fingers, to ground the abstract concept of quantity in something array from more basic mathematical structures [15]. physical. In middle school and high school, graphs are relied upon heavily to aid in the understanding of slope as rise over Drawing inspiration from the graphics in Mathematics of run or representing integrals as the area under a curve. Even in Big Data, this project created a similar representations of undergraduate years, visual intuition still arises. For example, abstract structures while increasing the scope to a large group in linear algebra, matrix multiplication is represented by linear of fundamental structures in abstract mathematics. This project transformations. However, as mathematics students move from produced a catalog of different maps representing the connec- tions between abstract mathematical objects. It also categorizes This material is based in part upon work supported by the NSF under objects based on properties, gives a list of axioms sufficient grant number DMS-1312831. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do to constructing each structure and includes other features that not necessarily reflect the views of the National Science Foundation. make the map aesthetically appealing. II.BACKGROUNDAND DEFINITIONS Types Informally, a mathematical obejct is any structure built from Previously defined objects. existing mathematical objects and structures, beginning with Functions sets as the basic building block. Matrices, functions, relations, Described functions (Definition 2.2) that must be and sets are all examples of mathematical objects. The terms defined over the structure. “object” and “structure” will be used interchangeably. For Relations clarity, we define what is meant by certain terms central to Enumerated the relations ((Definition 2.1) that must the project: be defined over the structure. Definition 2.1: Relation An n-ary relation R between sets Properties Detailed additional properties that functions and re- X1,...,Xn is a subset of X1 Xn × · · · × lations over the structure must satisfy. Definition 2.2: Function An n-ary function f : X0 X Y is an (n + 1)-ary relation f X X× · · ·Y × For ease of presentation, the functions and relations on the n → ⊆ 0 × · · · × n × where for each x0 X0, . . . , xn Xn there exists a unique structures were not written in terms of set theoretic functions y Y such that (x ∈, . . . , x , y) ∈f. Such unique y is denoted (e.g. union, intersection) and relations (e.g. membership). ∈ 0 n ∈ by f(x0, . . . , xn) Although set theoretic definitions could be written in principle to ensure that the functions and relations over structures III.APPROACH really behaved as they claimed to, it would be cumbersome The first step was to survey pure mathematics as a whole, and hinder the definitions that are used more commonly in selecting objects that were central enough to the entire field literature and most fields of mathematics. that most pure mathematicians would know them, excluding Many structures included in the diagrams have multiple categories, which were too board to fall within the scope of the accepted and useful definitions. In most cases, it did not make project. The research involved a broad range of textbooks and sense to list more than one definition, since there was not a internet resources [15]–[19]. The next step was to determine consistent or elegant way to do so. In general, the definition a hierarchical relationship between all of the structures. The that would be presented earliest to a student learning in a hierarchical relationships were created by determining the traditional classroom. However, often the relationship between amount of structure encoded by the axioms of an object. For structures it was related to would be defined normally in terms example, there is more structure encoded in the axioms of a of an equivalent definition, so for consistency, the relationship module than those of a group. would need to be modified to be in terms of the definition The overall layout of the objects were facilitated with the stated within the node. The only exception depicting on the concept map program CMAP [20]. CMAP allows for rapid maps were Lattices, which have an algebraic and an order manipulation of visuals. Nodes were written to represent each theory defintion. Niether could be excluded because many object and labeled arrows were drawn between the nodes of the different types of lattices, both order theoretic and to represent their relationship to each other. Although all algebraic properties of the lattice were used in definitions. mathematical objects are related to all other mathematical While some structures have multiple accepted standard objects, drawing most connections would be redundant and definitions, there are other structures that have no standard clutter the diagram. Therefore, the connections that were definition. For example, some but not all authors require that drawin on maps were unique (with some exceptions disccused rings have a multiplicative identity. To deal with nonstandard later in the paper) Much of the thought of how the connections definitions, as few choices for definitions were made as should be drawn and what the placement of each structure possible. Although it would be possible to include multiple should encode was done at this stage. definitions, it was decided that it would quickly become The objects were next enumerated in LaTeX as boxes and cumbersome. When choices in definitions were made, they given descriptions that were rigorous and as complete as pos- were made to follow the what the majority of authors defined sible while preserving a high degree of simplicity. Information the structure as. Distinctions generally only needed to be made about each structure was entered in its corresponding node in at a low level of complexity, as structures that built upon a uniform and consistent manner. Using TikZ [21], each node them only needed to reference the structure as a whole, and was positioned using the placement determined in the CMAP not the internal mechanisms of the structure. Additionally, the stage. Also, with the assistance of TikZ, arrows were drawn nuances of operations were not stated in the diagrams, which between relevant structures and labeled with brief descriptions is where most of subtleties involving non standard definitions of the relationship between the structures. The overall map was applied. For example, a module is an abelian group that is also shaded so that it would be easier for an individual to use. closed under ring multiplication. But the mechanism of scalar multiplication is not discussed, which is where it is determined IV. DESIGN what properties a ring of scalars must satisfy, so there is no Balancing concise and complete information in the diagrams conflict. was very important for user readability. There were four Precedence was given to simplicity and intuitive representa- main categories that all characteristics of an object could be tion. Therefore, certain technically unnecessary characteristics described as: of structures were only listed if they gave a significantly fuller Riemann Manifold Types: A smooth manifold M 2 Functions: gp TpM R Relations: Properties: gp ∶is the inner→ product on the tangent space of a point p gp X,Y gp Y,X gp aX Y,Z agp X,Z gp Y,Z ( ) = ( ) M’s metric ( + ) = ( ) + ( ) tensor is now positive definate

Forest Any point on M Pseudo Riemann Manifold has a positive def- Types: An undirected graph G V,E Types: A smooth manifold M Functions: inite metric tensor 2 Functions: gp TpM R Relations: = ( ) picture of the structure. For example, when describing a lie Relations: every point Properties: G’s connected component are trees, but G need not be connected Properties: g ∶is the inner→ product on the tangent space of a point p on M has a p g X,Y 0 Y X 0 Smooth Manifold non degenerate p Types: A manifold M metric tensor Functions: ( ) = ∀ ￿⇒ = Relations: A collec- Properties: Derivatives of arbitrary orders exist tion of trees

Line Graph Types: An undirected graph L G V ,E , with underlying undirected graph G V,E Bipartite Graph M is infintiely Types: A k-partite graph G V,E Functions: f E V ∗ ∗ Edges represent Tree di↵erentiable ( ) = ( ) = ( ) Functions: Relations: x, y∗ E vertces (and Types: A undirectd graph G V,E ∶ → Relations: = ( ) Properties: f∗maps edges in∗ G to vertexes in L G visa versa) of Functions: ∼ = {{ } ∈ }￿ Properties: Special case of k-partite graph for k 2 An unordered pair of vertices vi ,vj E i↵ their corresponding edges in G share an endpoint another graph, G Relations: = ( ) Manifold ( ) V ,V are disjoint sets partition G’s vertices ∗ ∗ ∗ Properties: There are no cycles in G 1 2 Types: A topological space, M { } ∈ All e E can be written as v ,v for= some v V and v inV G is connected i i j i 1 j 2 Functions: G has no odd cycles Compact Manifold Relations: G is 2-colorable∈ { } ∈ M is compact ?? Types: A manifold M Properties: p M, u p u n as a space R A special vertex Functions: Manifolds are not inherently metric or inner product spaces is called the root Relations: each point locally ∀ ∈ ∃ ∈ U( ) ∶ ≅ Properties: X x C x, then for F C, F finite, X x F x ’looks like’ G can be prti- tioned into ∈ ∈ euclidean space = ￿ ⊆ = ￿ Rooted Tree G is acyclic Hypercube Graph exactly 2 sets Types: A tree G and connected Types: A regular, bipartite graph Qn X,E Functions: Functions: algebra, it is redundant to note that both and Relations: Relations: = ( ) n n 1 [X,X] = 0 Properties: One vertex is designated as the root Properties: Qn has 2 vertices and 2 n edges − K-partite Graph The children Types: A graph G V,E Euclidean Space of each node G is directed Functions: Types: A banach and hilbert space n R are ordered Relations: = ( ) Functions: Undirected graph H is the graph Properties: V ,V ,...,V are disjoint sets that partition G’s vertices Relations: Types: A graph G V,E 1 2 n of an n dimen- All ei E can be written as vi,vj where vi Vi,vj Vj and i j n 2 Functions: Properties: A norm is defined as x x, x i 1 xi Ordered Tree Arboresence and Antiarboresence G is directed sional cube G is k-colorable n Relations: = ( ) d x, y x y 2 ￿ ￿ Types: A directed graph G V,E Types: A rooted polytree, G V,E ∈ { } ∈ ∈ ≠ i 1 i i = Properties: is irreflexive and symmetric ￿ ￿ ￿ = ￿ ￿ = ∑ ( ) Functions: Functions: ( ) = ∑ = ( − ) Relations: = ( ) Relations: = ( ) ∼ Endow with an Endow with Properties: The children of nodes in G are ordered Properties: Every node has a directed path away from or toward the root inner product a norm Regular Graph Types: A graph G V,E Functions: There is a directed Relations: = ( ) path between Properties: All v V are connected to the same number of edges any vertex V can be di- Multigraph and the roote Hyper Graph ∈ vided into k Types: AtupleG V,E Types: AtupleH V,E pairs in E re- independant sets Functions: Functions: main unordered All vertexes have Relations: = ( ) Relations: = ( ) the same number More than 1 Properties: E is a multiset Hilbert Space Properties: E ei i Ie,ei V Banach Space Polytree of neighbors edge can exist More than one edge may connect the same two vertices Types: An inner product space H Ie is an index set for edges in H Types: A vector space V Types: A directed graph G V,E between the Functions: = { ￿ ∈ ⊆ } Functions: Functions: same two nodes Relations: Relations: Relations: = ( ) Properties: A norm may be defined as x x, x , but was included because both give insight Properties: for any Cauchy Sequence xn ,limx xn x 0 Properties: G’s underlying undirected graph is a tree [X,Y ] = [Y,X] for any Cauchy Squence xn ,lim￿x xn x 0 G is acyclic →∞ ￿ ￿ = ￿ ￿ Chordal Graph { } ￿ − ￿ = and connected { } →∞￿ − ￿ = Types: A perfectly orderable graph G V,E Archimedian Local Field Nonarchimedian Local Field elements in E may Functions: Types: A local field K Types: A local field K V is complete V is complete be any subset of V Relations: = ( ) as a space as a space Functions: Functions: Properties: All induced cycles are of order 3 Relations: Relations: The manifold G has a perfect elimination order Properties: The sequence n is unbounded Properties: The sequence n is bounded Any cycke of 4 or additionally has n 1 n 1 Simple Graph Perfect Graph more has a chord group structure ≥ ≥ Simple Directed Graph Types: AtupleG V,E Types: A graph G X,E Perfectly Orderable Graphs ￿ ￿ ￿ ￿ Chromatic number Inner Product Space Types: A graph G V,E Functions: Functions: ! G n Types: A perfect graph G V,E Normed Vector Space of every induced Types: A vector space V and field of scalars F Functions: pairs in E Relations: x, y =V( V) x, y are ajecent vertices in G G n = ( ) Functions: Types: A metric and topological vector space V 2 subgraph is G has a relation, Functions: , V F Relations: x, y =V( V ) x, y E are ordered Properties: V is the set of vertices in G Relations: ∶ ￿ Relations: x, y V V =x( y ) Functions: V R Absolute the size of the Relations: Absolute Value is a Properties: is irreflexive E∼=is{{ made} ∈ of two× element￿ subsets of V } Properties: ! ∶G ￿is the the size of the largest clique in G Properties: is a total order Relations: Value is not a endowed with largest clique < Properties: ￿x,⋅ ⋅￿ y∶ y,x→ bounded function u,= {( v V,u) ∈ v× v￿( u) ∈ } is the adjacency relation X is the the chromatic number of G For<= {( any chordless) ∈ × path￿ < abcd} G and a b,thend c Properties: ￿￿x ∶ 0→ for x 0 and x 0i↵ x 0 bounded function Graph Algebra a function to edges represent x, x 0 is irreflexive and symmetric !(G) G < ↵x ↵ x Types: A directed graph D V,E multiply vertices, comparable Comparability Graph ￿x, x￿ = ￿0i↵ ￿x 0 2 ∀ ∈ ⇏ ∼ ( ) ∈ < ￿ ￿x￿ >y x ≠ y ￿ ￿ = = Functions: V V 0 elements in a poset Types: A perfect graph C S, ￿ax, y￿ ≥ a x, y ∼ ( ) = ( ) ￿ induces￿ = ￿ ￿￿ a￿ metric on V d x, y x y Relations: = ( ) ⋅ There exists a Functions: ￿x y,z￿ = x, z= y,z ⋅∶ → { } = ( ⊥) ￿ + ￿ ≤ ￿ ￿ + ￿ ￿ Local Field xx,y￿ V, x, y E path between Relations: x, y S x y − ￿ ￿ = ￿ ￿ Properties: x y any two veritces Properties: S is the set of vertices ￿￿ ∶ ( ) = ￿ − ￿ ￿ + ￿ = ￿ ￿ + ￿ ￿ Types: A topological Field K ￿0 x, y V 0 , x, y E ￿ ∈ ( ) ∈ on the graph S⊥=is{( a poset) ∈ under￿ < } Functions: K R 0 0 ⋅ V∶ ￿ ordered pairs Complete Metric Space Relations: ￿ ∈ ￿ { } ( ) ∉ is the edge relation ≥ ￿ in E the struc- Types: A metric space M Topological Vector Space ￿￿∶ → x, y , y,z E < x, z E Properties: u, v K, , ⌧ u ,v , ∉ ture of a group Types: A vector space V over a topological field K Equipped with ⊥ Functions: if C ⌧ satisfies X x C x, then there exists F C, F finite X x F x Connected Graph equipped with Functions: V 2 V a function , ( ) ( ) ∈ ￿⇒ ( ) ∈ Relations: ∀ ∈ ∃U V ∈ ∶ ∈ U ∈ V U ∩ V = ￿ Types: A graph G V,E a function k V V ∈ ∈ Properties: F U A F A A U, F converges ⊆ = ￿ ⊆ = ￿ Cayley Graph Functions: d x, y x y + ∶1 V→ V ￿ ￿ ￿￿ Types: A directed graph G, S , S a generating set for group G Relations: = ( ) ∀ ∀ ∶∃ ∈ ∶ × ⊆ Relations: −× → Each vertex has ( ) = ￿ − ￿ + Functions: f G V G Properties: There exists a path between any two v V in the graph Properties: k− V∶ represents→ scalar multiplication by elements in a field K K is a field of a unique label 1 g S CS = ( ) Each edge has , and scalar multiplication are continuous scalars for V Has an absolute Relations: E∶ → (g,gs) G G g G, s S a unique label ∈ to the nature of a lie bracket. × − ∶ → + −+ value function Properties: f assigns each g G to a vertex in g (assigns) = {( each)s∈ S×to a￿ ∀ unique∈ color∀ ∈ } ∈ M is complete as a add a mean- Topological Field The identity element generally is ignored, so the graph does not contain loops Vertex Labeled Graph Strict Totally Ordered Set ∈ topological space ingful origin Types: A field K with topology ⌧ Types: A graph G V,E Types: A poset P 2 Functions: K K x, y x y Functions: f V L Functions: 2 Edge Labeled Graph K K x, y x y Relations: = ( ) Relations: x, y P P x y ⋅∶ 1 → ∶ ( ) ￿ ⋅ Types: A graph G V,E and set L K R Properties: f ∶maps→ vertexes to a set of labels L Properties: is irreflexive, transitive, asymmetric and trichotomous Metric Space 1 Functions: f E L Ane Space +−∶ K→ 0 ∶ ( K ) 0￿ + <= {( ) ∈ × ￿ < } Types: A topological space M + Relations: = ( ) Relations: −− ∶ → < 2 Types: A set A and vector space A × ∶ → Functions: d M R Properties: −, , ∶ 1￿,{ }1 →are continuous￿{ } mappings Properties: f maps edges to a set of labels L Functions: A A A a, v a v Finite Countable Set Relations: ￿→ − − all elements in P + ⋅ Types: A set S becomes Properties: d ∶x, y →0 Relations: ￿→ + ⋅ − − Functions: are comparable irreflexive d x, y 0 x y Properties: A×’s additive→ ∶ ( group) ￿ acts+ freely and transitively on A has inverses Relations: ≤ d(x, y) ≥ d y,x S is in bijec- Uncountable Set ￿→ v,w A, a A, a w u a w u in the set Properties: S d(x, z) = d ⇔x, y = d y,z 0 tion with a Types: A set S edge labels are a, b A,￿→ v R b a v ⋅ ( ) = ( ) ∀ ∈ ∀ ∈ ( + ) + = + ( + ) subset of the Functions: comparable ( ) ≤ ( ) + ( ) a A, A A￿→ v a v is bijective ￿ ￿ < ℵ ∀ ∈ ∃ ∈ ∶ = + natural numbers Relations: ￿→ Properties: S 0 ∀ ∈ → ∶ ￿ + Strict Poset Totally Ordered Set Closed sets are ￿ ￿ > ℵ Types: A poset P Types: A poset P those who are the Countable Set Weighted Graph Functions: Functions: common zeros of Relations: x, y P 2 x y Relations: Topological Ring Types: A set S There is no in- Types: An edge labeled graph G V,E and labels L a subset of the Properties: is irreflexive, transitive, and asymmetric Properties: All elements are comparable under Types: AringR with topology ⌧ Functions: jection from S Functions: indeterminants 2 <= {( ) ∈ ￿ < } Functions: R R x, y x y Relations: into the nat- Relations: = ( ) in a polynomial 2 < ≤ Closeness relation R R x, y x y Properties: S 0 ural numbers Properties: L is an ordered set ring over the same 1 is replaced by ⋅∶ R→ R∶ ( ) ￿ ⋅ Complete Uniform Space underlying field Relations: +−∶ → ∶ ( ) ￿ + ￿ ￿ ≤ ℵ distance fucntion d + Types: A uniform Space U Properties: −distributes∶ → over 1 Infinite Countable Set S is in bijec- all elements in P Functions: , , are continuous mappings is now irreflexive Relations: ⋅ − + Types: A set S tion with the are comparable + Although there are many fundamental relationships between Properties: F U A F A A U, F converges + ⋅ − Functions: natural numbers ≤ Relations: Graded Poset ∀ ∀ ∶∃ ∈ ∶ × ⊆ Properties: S Partially Ordered Space 0 Types: A poset P bijection into N S has an injec- Types: A poset X endowed with topology ⌧ U is complete as a Functions: ⇢ P N ￿ ￿ = ℵ tion into the Functions: topological space Relations: x, y P z x z y natural numbers Relations: x, y X X x y Properties: x ∶ y → ⇢ x ⇢ y Uniform Space add a new opera- Properties: x, y X,x y , ⌧,x ,y u v, u , v x￿= {(y ) ∈⇢ y￿ ￿ ⇢∶ x< <1 Types: A topological space U and entourages Topological Space tion + so G, ≤= {( ) ∈ × ￿ ≤ } < ￿⇒ ( ) < ( ) Functions: Types: A set X and ⌧ X is a group and ∀ ∈ ￿ ∶∃U V ⊂ ∈ U ∈ V ∶ ￿ ∀ ∈ U ∀ ∈ V ￿ ￿⇒ ( ) = ( ) + Relations: U x, y x y Functions: G, is a moniod( +) P endowed u Relations in P has a rank Properties: is a nonempty collection of relations on S Relations: ⊆ ℘( ) Lie Group with a topology define closeness function, ⇢ x U = {(and)U￿ ≈ V } X X V Properties: X, ⌧ ( ⋅) Types: A topological group G that respects U , V , x, y, z V V x V y,y V z x U z xi ⌧ i I xi ⌧ for finite I X has a group Functions: 1 ( ) U∈ , V ⊆, ⊆x, y ×V V￿⇒ y∈ V x x U y xi ￿⌧∈ xj ⌧ structure with con- Topological Group , are smooth Relations: ≤ j∈J Preordered Set Partially Ordered Set U,∀ V∈ ∃ ∈ U∀ V ∶ ○ ⊆ ∶ ≈ ≈ ￿⇒ ≈ ∈ ￿⇒ ￿ ∈ tinuous functions Types: A group G with topology ⌧ maps,− the under- Properties: p G, p Br p , for some r 0 Set Locally Finite Poset ∈ Types: A set P Types: A set P ∀ ∈ ∃ ∈ ∀ ∶ ○ ⊆ ∶ ≈ ￿⇒ ≈ ∈ ￿⇒ ￿ ∈ Functions: G2 G lying⋅ − topology is a 1 and are smooth maps Types: a collection of objects any subset of Types: A poset P equipped with Functions: Functions: ∈ ￿⇒ ∩ ∈ 1 G G smooth manifold ∀− ∈ ∃U ≅ ( ) > Functions: , , , , , , is antisymmetric P has finitely Functions: a preorder, Relations: x, y P P x y Relations: x, y P P x y 1⋅∶− G → − × Relations: many elements Relations: − ∶ → ⊂ ∩ ∪ ￿ × ℘ ￿￿ Properties: All elements need not be comparable under ≤ Properties: All elements need not be comparable under Relations: Properties: ≤ ≤= {( ) ∈ × ￿ ≤ } ≤= {( ) ∈ × ￿ ≤ } Properties: x, y P,x y x, y has finitely many elements ∈ 1 ∈ transitive and reflexive transitive, reflexive and antisymmteric Properties: , are continuous mappings ≤ P has group ≤ ∀ ∈ ≤ ∶ [ ] − ≤ structure that ≤ ⋅ − respects each element ≤ is the greatest lower bound of S is equipped members of S are Neighborhoods of Partially Ordered group the compact with a relation sets but S is not points must satisfy Types: A poset and group G elements below it Functions: G2 G stronger properties → 1 G G 0+−∶ G → different mathematical structures (e.g. the association of a Lie + any two ele- Relations: − ∶ x,→ y G G 0 x y Convergent Space ∈ ments have an Proper Class Properties: respects Types: A set S and filters on S, Cardinal ≤= {( ) ∈ × ￿ ≤ − + } upper bound Algebraic Poset Types: a collection, C of sets The set of ordinals Functions: Types: Ordinals  ≤ + Types: A poset P Functions: ' x that cannot be put Relations: F Functions: Functions: Relations: in bijection with Upward (Downward) Linked Set Properties: F ,x S, F x F convergences to X ( ) Relations: Types: A subset S of poset P Relations: → Properties: ' x is a membership function any lesser ordinal is centered, isotone, and directed Properties: The cardinality  is the least ordinal, ↵, s.t. there is a bijection between S and ↵ Functions: Properties: each element is the least upper bound of the compact elements below it ∀ ∈ F ∈ → ⇔ C is not a set x F F x ( ) Relations: → Properties: x, y S z P s.t. x z and y z N ( ) = ￿{ ∈ F￿ → } ∀ ∈ ∶∃ ∈ ≤ (≥) ≤ (≥) Each element any two elements is either the set of P have an Magma of the elements upper bound Types: A set M before it or 0 Functions: M 2 M Relations: M is cancellitive is associative Upwards (Downwards) Centered Set Properties: ⋅∶u, v →M u v M Ordinal Types: A linked subset S of poset P ⋅ Types: A collection of sets, Xi Functions: Quotient Ring ∀ ∈ ∶ ⋅ ∈ n Functions: , ,X ,S n Relations: Types: AringQ Quasigroup Relations: Functions: Properties: Z P,Z has an upper (lower) bound P Semigroup Types: A magma Q Properties: X× k+ 0, 1,(2,...X) k 1 Relations: x, y R R x y I , 1 are contin- Types: A magma S of elements Functions: Q2 Q > is trichotomous, transitive, and wellfounded Prime Ideal − ∀ ⊆ ∈ Properties: Q is constructed by dividing R with one of its ideals 2 uous− mappings Functions: Q Q = { } ∼= {( } ∈ × ￿ − ∈ } Types: A subgroup of elements I ⋅ − Relations: 1￿R∶, 1L →Q > any subset Functions: S is endowed with Properties: is associative Relations: ￿∶ → of P has an Relations: Semiring a new operation, Properties: Q is a∈ cancellative magma Properties: ab I a I or b I Types: A set R upper bound such that ⋅ Functions: R2 R distributes over group with its corresponding Lie algebra), for simplicity and ∈ ￿⇒ ∈ ∈ 2 Q has an iden- R R + ⋅ Q has an iden- 0⋅∶ R → + tity element, 1 If the product of tity element, 1 Upward (Downward) Directed Set A ring formed Relations: + → two elements is Types: A preset P by the quotient Properties: ∈is a commutative monoid in I, then one of Functions: of a ring and is a monoid and distributes over Monoid Loop Maximal ideal 1 one of its ideals those, elements, are + Types: A semigroup S Types: A quasigroup Q Relations: Types: A subgroup of elements, I continuousmust− be in I Principal ideal ⋅ + Functions: 1 S Functions: 1 Q Properties: is a preorder Functions: + + − ⋅ Types: A subgroup of elements, I Relations: Relations: x, y P z s.t. x z and y z Relations: ≤ Functions: P Group Properties: 1∈ is the identity for Properties: 1∈ is the identity for Properties: I J I J or J R ∀ ∈ ∶∃ ≤ (≥) ≤ (≥) Relations: Types: A group G 1 u u 1 u 1 x x 1 x Properties: I is generated by a single element ⋅ ⋅ ⊆ ￿⇒ = = Functions: Relations: ⋅ = ⋅ = ⋅ = ⋅ = I is not the proper Properties: g G g pn,p prime, n subset of any N other ideals of R I is generated from ∀ ∈ ∶ ￿ ￿ = ∈ a single element Torsion Free Group Ideal any pair of any pair of all elements Types: A group G Types: A subgroup of a ring, I elements have a elements have in S have an is associative Functions: Functions: unique greatest a unique least inverse in S Relations: Relations: n lower bound upper bound I is a right and left ⋅ Properties: g G, g 1 g 1,n N Free Abelian Group Left (right) Ideal Properties: The left and right ideals generated by a subgroup are equal idea of the same all elements Types: A group F Types: A subgroup of a ring, I x I, r R rx I n underlying subset have order p ￿ ∈ ≠ ∶ = ∈ Functions: Functions: Relations: Relations: ∀ ∈ ∀ ∈ ∶ ∈ there are no add commuta- Properties: F is generated by a set S Properties: x I, r R rx xr I elements in G tivity relation The only relation on F is commutativity Sigma Algebra ∀ ∈ ∀ ∈ ∶ ( ) ∈ of finite order Types: A subset ⌃, of X , X a set Free Group A subset of the Topological group Functions: , , Group Types: A group F ?? ?? ring closed under ℘( ) Types:AbelianA Groupgroup G Relations: Types: A set G Functions: Types: A poset P Types: A poset P to avoid determining what constitutes being ‘fundamental’ left (right) ring Functions: 2 Properties: X∩ ∪ ⌃￿ Types: A group G Functions: G G There are no fur- Relations: Functions: Functions: multiplication , closed under countable operations Relations:Functions: 1 G G 1 is commutative ther relations on G Properties: F is generated by a set S Relations: x, y P P x y Relations: x, y P P x y R, be- ∈ Properties:Relations: and are continuous maps ⋅∶ → A subset of the 1 − G There are no relations on FS (beyond group axioms) Properties: x, y S, c c x, c y Properties: x, y P, c x c, y c comes a group ∩ ∪ Properties:− is commutative − ∶ → ring closed under ⋅ Relations: c≤=is{( the unique) ∈ × greatest￿ ≤ lower} bound c≤=is{( the unique) ∈ × least￿ upper≤ } bound ( +) − × ring multiplication A ring under Properties: is∈ associative ∀ ∈ ∃ ∶ ≤ ≤ ∀ ∈ ∃ ∶ ≤ ≤ symmetric ⋅ 1 is the identity for Meet Semilattice: Algebraic Edition Join Semilattice: Algebraic Edition di↵erence as and ⋅ 1 is the inverse map for G has a fi- Types: A poset P Types: A poset P 1 1 2 2 , , , set intersection as − ⋅ nite number Functions: P P Functions: P P are continu-− − + − ⋅ of elements Relations: Relations: Noncommutative Ring + ⋅ ous+ ⋅ mappings− − ⋅ G has no nor- Properties: ∧∶is associative,→ commutative, and idempotent Properties: ∨∶is associative,→ commutative, and idempotent Types: AringR mal subgroups Functions: ∧ ∨ Relations: Properties: need not be commutative Boolean Ring any pair of Artinian Ring ⋅ Simple Group elements have Types: A commutative ring R Types: AringR Commutative Ring Types: A group G a unique least All elemets are Functions: Functions: Functions: upper bound Types: AringR Relations: G has a fi- Finite Group Relations: idempotent with Relations: Functions: respect to , Properties: x R x2 x nite number Types: A group G Permutation Group Properties: For any I0 I1 ... Ik there is some k 0 s.t. Ik Ik 1 Relations: Properties: N G N 1 or G of generators x R 2x 0 Functions: Types: A group Pn + Properties: is commutative ∀ ∈ ∶ = Relations: ⊇ ⊇ ⊇ ≥ = ⋅ + ◁ ￿⇒ = { } Functions: ∀ ∈ ∶ = Properties: G n, n N Relations: ⋅ Properties: Pn Sn ￿ ￿ = ∈ need not be G is closed under P contains only < n commutative multiplication by even permutations ⋅ a ring of scalars Semimodule Finitely Generated Group A proper subset Noetherian Ring Finitely Presented Group Types: A set M Types: A group G G is the group of the Finite Types: AringR 2 Types: A finitely generated group G Functions: M M Functions: G has fi- of all bijective Symmetric Group Functions: Functions: R M M Relations: nite relators functions from Alternating Group in general, only associations which indicate one structure Relations: Ideals in R satisfy + ∶ → Relations: Relations: Properties: G is of the form S R , S a generating set, R relators the set to itself Types: A permutation group An Properties: For any I0 I1 ... Ik there is some k 0 s.t. Ik Ik 1 the descending Properties: is× associative→ and commutative Properties: R n, n N S n, n N Fintie Symmteric Group Functions: chain condition M, is an commutative monoid + ￿ ￿ ￿ Types: A finite group S Relations: ⊆ ⊆ ⊆ ≥ = R+ M is scalar multiplication by elements in a ring ￿ ￿ = ∈ n is additionally ￿ ￿ = ∈ Functions: Properties: An Sn even Scalar( +) multiplication is associative and distributes over addition commutative Relations: ⋅ × = { ￿ ∈ ∧ } Ideals in R satisfy Properties: Sn X X is bijective the acending M is a = { ￿ ∶ → } chain condition Ring abelian group No nontriv- Types: A set R ial partitions Functions: R2 R Artinian Module G has one are preserved R2 R Types: A module M generator There are no ⋅∶ 1 → R R Functions: Primitive Group zero divsors in R + ∶ → 0 − R Relations: Types: A permutation group G that acts on a set X −+ ∶ → Relations: Properties: For any chain of submodules S0 S1 ... Sk there is some k 0 s.t. Sk Sk 1 Functions: Domain Properties: R,∈ is an abelian group + Relations: Types: AringR ⊇ ⊇ ⊇ ≥ = M satisfies is a monoid and distributes over R is a ring of Properties: G preserves only trivial partitions Functions: ( +) decending chain scalars for an S is also a join S is also a meet Relations: ⋅ + condition on abelian group M Module semilattice semilattice is commutative Properties: a, b R, ab 0 a 0 or b 0 submodules Cyclic Group Types: A set M and obeys ab- and obeys ab- Semimodular Lattice Functions: M 2 M Types: A group G with element a Distributive Lattice sorption law sorption law ⋅ ∀ ∈ = ￿⇒ = = Modular Lattice Types: A lattice L Noetherian Module 1 M M Cyclically Ordered Group Functions: G has a chain of Polycyclic Group Types: A modular lattice L L has a re- M satisfies L’s functions Types: A semimodular lattice L Functions: Types: A module M R+−∶ M → R Types: A group C Add a rela- Relations: cyclic subgroup Types: A group P with subgroups H Functions: stricted asso- accending chain + i distribute over Functions: Relations: x, y P z s.t. x z y Functions: Relations: − ∶ → Functions: tion, ,, Properties: G can be generated from a and a set of relations R whose quotients Functions: Relations: ciative property } condition on each other Relations: Properties: a b a b a b Relations: Properties: is× associative→ and commutative Relations: ,, a, b, c C C C a, b, c G is necessarily commutative are also cyclic Relations: Properties: x y z x y x z ￿= {( ) ∈ ￿ ￿ < < submodules Properties: a c a b c a b c Properties: For any chain of submodules S S ... S there is some k 0 s.t. S S M, is an abelian group Properties: a, b, c is cyclic, asymmetric, [ ] Properties: P is necessarily finitely presented x y z x y x z ∧ ￿ ￿⇒ ￿ ∨ 0 1 k k k 1 L is addition- All submodules of M are finitely generated R+ M is scalar multiplication by elements in a ring transitive[ ] = {( and total) ∈ × × ￿[ ]} H H H is cyclic ∧ ( ∨ ) = ( ∧ ) ∨ ( ∧ ) Integral Domain + All finitely gen- i i 1,2,..,n 1 i i 1 ≤ ￿⇒ ∨ ( ∧ ) = ( ∨ ) ∧ ⊆ ⊆ ⊆ ≥ = Scalar( +) multiplication is associative and distributes over addition [ ] G is a subgroup in ∨ ( ∧ ) = ( ∨ ) ∧ ( ∨ ) ally algebraic Types: A domain R erated subgroups ∈{ − } + × another group V { } ∶ ￿ and atomic extending another by the additional of new functions, relations, Functions: of G are cyclic Relations: Geometric Lattice Properties: is additionally commutative n 2 Add a functon, M’s ring is di↵er- Types: A semimodular, algebraic, atmoic lattice L ential operators Heyting Algebra ⋅ ≤ ￿→ Functions: Types: A distributive lattice H Relations: 2 Virtually Cyclic Group Functions: H H Properties: L is finite Locally Cyclic Group Metacyclic Group Types: A group V with subgroup H Relations: Types: A group L Types: A group G ￿→ ∶ → Ring addition is Functions: Properties: c a b c a b All elements Functions: Functions: the symmetric Relations: Monoid H, , 1 has operation can be factored Relations: Relations: ( ∧ ) ≤ ⇔ ≤ ( ￿→ ) If a covers the di↵erence and Properties: H V H is cyclic into the product Properties: H L,ifH is finitely generated, it is cyclic Properties: n 2 ( ⋅ ) ⋅ = ∧ greatest lower L is additionally M has no ring multiplica- V H n, n N algebraic and of a unit and M is generated bound of a and zero divisors tion is greatest ∃ ≤ ∶ Atomic Lattice semimodular prime elements DModule from a finite set ∀ ≤ ≤ b,thenb is lower bound ￿ ∶ ￿ = ∈ a b a b Types: A lattice L Types: A module M X covered by the Functions: Functions: @ M M greatest lower Ri ⋅ = ∧ Relations: x, y P z x z y Relations: [ ] bound of a and b Finitely Generated Module Properties: b L 0 b 0 } L is additionally Properties: X ∶ x ,x→ ,...,x where x is an indeterminant H’s underlying 1 2 n i Types: A module M ￿=a {(L 0) ∈ a ￿ ￿ ∶ < < semimodular @ is in the ring of scalar multiplication lattice is complete i i Ri Functions: ∀b ∈ a ∶, a∧ 0= a b and atomic @ = satisfies{ the Leibniz} product rule i i i Ri Relations: ∃ ∈ ∶ ￿ M’s generating Simple Module ai i I is the set of atoms in L Properties: M has a finite number of generators set is one element ∀ ≠ ∃ ∶ < < Types: A module M Residuated Lattice ∈ Unique Factorization Domain { } Types: A lattice L Types: An integral domain R Torsion Free Module Cyclic Module Functions: Complete Heyting Algebra M has no proper Functions: L2 L Functions: Types: A module M Types: A finitely generated module M Relations: Types: A heyting algebra H submodules 1 L Relations: Functions: Functions: Properties: The only submodules of M are 0 and M Functions: L only must satisfy Relations: ⋅∶ → Properties: All ideals are finitely generated Relations: Relations: Relations: the Modular law Properties: L,∈ , 1 is a monoid All irreducible elements are prime Properties: m M,r,m 0 r m 0 Properties: M is generated from a single element Properties: H is complete as a lattice in the special L, is a lattice case b a x R, x 0,x up1p2...pn for u unit, pi prime ( ⋅ ) All non zero ele- ￿ ∈ ≠ ∶ ⋅ = z x, there exists a greatest y and y, there exists a greatest x s.t. x y z ⊥ Algebraic Lattice ( ≤) ments are either ∀ ∈ ≠ = = Types: A continuous lattice A ∀ ∶∀ ∀ ⋅ ≤ atoms or compa- or properties were included. The inclusion of examples of rable to atoms Functions: Relations: Properties: Each element is the least of compact elements below it: those x who satisfy x a

All ideals are ￿ principal Lattice (order theory) Types: A set L Functions: x y L has an element each are the great- 1 and a function Relations: x, y L x y est lower bound Principal Ideal Domain ￿→ = ¬ ∨ such that L, , 1 Properties: x, y L, c, d c is the greatest lower bound and d is the least upper bound of the compact ≤= {( ) ∈ × ￿ ≤ } Types: A unique factorization domain R is a monoid ⋅ elements below it Functions: ( ⋅ ) ∀ ∈ ∃ ∶ Relations: Properties: All ideals are principal Any two elements have a greatet common divisor Relational Algebra M has no re- Types: A commutative algebra R Complete Lattice lators beyind Functions: R2 R Types: A lattice L module axioms R2 R Functions: ∨∶R →R Relations: Relations: ∧∶ → Properties: A L, A has a greatest lower bound and least upper bound Properties: −∶ → Lattice (Algebra) R can be en- Types: A set L ∀ ⊆ dowed with a eu- Functions: L2 L 2 clidean algorithm Orthomodular Lattice L L Relations: ∨∶ → Types: A lattice L a is the least upper Properties: ∧∶L, ,→L, are both semilattices connected by absorption laws Lie Algebra Functions: bound for the a, b L a a b a, a a b a Euclidean Domain Types: A nonassociative Algebra g :( Relations: set of elements Exponential Field ( ∨) ( ∧) Types: A principal idea domain R Functions: g2 g Properties: a b b a b a way below a Types: A field K ∀ ∈ ∶ ∨ ( ∧ ) = ∧ ( ∨ ) = Functions: fi R 0 R Relations: ⊥ structures was on a case-by-case basis, due in large part to Functions: K K []∶ → ≤ ￿⇒ = ∨ ( ∧ ) Relations: Properties: x, y g x, y y,x ∶ ￿ { } → Relations: + × Properties: fi is the euclidean (gcf) algorithim ∶ → x, y, z g x, y,z y z,x z, x, y 0 Polynomial Algebra Properties: is a homomorphism ∀ ∈ ∶ [ ] = −[ ] R may have many fi or just one Di↵erential Field x, y, z g ax by, z a x, z b y,z and x, ay bz a x, y b x, z Types: A commutative Algebra A X Free Module ∀x, x ∈0 ∶ [ [ ]] + [ [ ]] + [ [ ]] = Functions: 1 A X Types: A polynomial field K every subset in Continuous Lattice The euclidean Types: A projective module M ∀ ∈ ∶ [ + ] = [ ] + [ ] [ + ] = [ ] + [ ] Relations: [ ] Functions: @i K K L has a least Types: A complete lattice A algorithm is one there exists a Functions: [ ] = Properties: X∈ [x ,x] ,...x where x is an indeterminate, n 1 Boolean Algebra L has a greatest Relations: 1 2 n i upper bound Functions: for all values homomorphism Relations: Types: A complemented, distributive lattice and algebra B lower bound and Properties: Finitely∶ → many @i exist over K All elements and a greatest Relations: x, y L L D L y sup D, x L, d D s.t. x d in the field from the addivitive Properties: M has no further relations beyond module axioms = { } ≥ Functions: B B least upper bound K has finitely @i satisfies the Leibniz product rule have relative lower bound Properties: x P, a a x is directed and has least upper bound x to multiplica- M has a basis B2 B L satisfies a a complements ￿= {( ) ∈ × ￿ ∀ ⊆ ∶ ≤ ∃ ∈ ∃ ∈ ≤ } tive groups many dera- @i is a homomorphism of K eqippued with A has an alphabet ¬ ∶ → Flexible Algebra Relations: c c for a c⊥ ∀ ∈ { ￿ ￿ } vations on it + a function , of indeterminates ⊕∶ → 2 Types: A nonasscoative Algebra A Properties: x B,x x ∨ ( ∧ Functions: is negation (complementation) = ≤ [ ] ∀ ∈ = Relations: x y x y x y ¬ L additionally x yx xy x Properties: x, y A x yx xy x is algebra addition, is algebra multiplication ⊕ = ( ∨ ) ∧ ¬( ∧ ) satisfies x y Relatively Complemented Lattice All elements y x y ′ ′ Types: A lattice L ( ) = ( ) ∀ ∈ ∶ ( ) = ( ) ⊕ ∧ are idempotent (′ ∧ ′ ) = Functions: ∨ ( ∧ ) Relations: Field Power Associative Algebra Properties: c d c, c, d a c, d , b s.t.: Types: AringK Types: A nonassociative Algebra A Commutative Algebra Orthocomplemented Lattice a b d Bounded Lattice Functions: K2 K Functions: Types: An algebra A a∀ ∀b ≥c [ ] ∶∀ ∈ [ ] ∃ 2 Indeterminates Types: A lattice L Types: A lattice L Characteristic Zero Field The multiplica- K K Relations: x x xx Functions: a ∨and=b are relative complements 1 in M are basis Nonassocative Algebra Alternative Algebra Functions: L L Functions: Types: A field K tive identity + ∶ K→ K Properties: x, y A x x xx xx xx x xx x xx xx Relations: ∧ = 1 vectors for V (x(xx))x= Types: An algebra A Types: A nonassocative Algebra A Relations: Relations: Functions: cannot be added ×∶− K→ 0 K 0 Properties: is commutative + ( )( ) = Functions: xx y x xy Functions: Properties: ⊥∶maps→ an element a to its complement Properties: x L Relations: finitely many 0−,−1 ∶ K → ∀ ∈ ∶ ( ( )) = ( )( ) = ( ( )) A eqipped with −× ∶ ￿{ } → ￿{ } ( ( )) Relations: Relations: a a 1 x 1 x Properties: n N 1 1 ... 1 0 times tog et the Relations: ( ) = ( ) ⋅ finitely many ⊥ ∀ ∈ ∶ ∈ Properties: is need not be associative Properties: x, y A xx y x xy a a⊥ 0 x 1 1 n additive identity Properties: +, both associate and commute derivations ∨ = L itself can act for each element ∧ = ￿ ∈ ∶ + + + = a ⊥ a x 0 x distributes over + ⋅ ∀ ∈ ∶ ( ) = ( ) ∧ = as the interval in L,thereisa ∨ = ￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿ × a ⊥ b⊥ a b x 0 0 0 is the identity for and 1 is the identity for K acts as scalars ( ) = second element ∨ = equipped with a × ⊥ ⊥ Complements the subtlety of distinguishing between a class of examples K, and K, are groups over a free module ≤ ￿⇒ ≥ in L such that ∧ = partial order, + × is commutative Di↵erential Algebra are unique glb and the lup ( +) ( ×) need not be Types: A commutative algebra A are the maxium ≤ assocative ⋅ Functions: @i A A Complemented Lattice and minimum Ordered Field ⋅ Relations: Types: A lattice L element of L Types: A characteristic zero field K ∶ → Properties: There are finitely many @i over A Functions: Functions: All @i satisfy the Leibniz product rule Relations: Relations: xy xx @i is a homomorphism of A Properties: a L, b L Properties: respects and Vector Space x y xx ( )( ) = + a b 1 K≤ is necessarily infinite Types: A set V over a field K Jordan Algebra ( ( ( )) a∀ ∈b 0∃ ∈ ∶ ≤ + × Functions: V 2 V The multiplica- Types: A nonassociative algebra A ∨ = Extension such 1 V V tive identity Functions: ∧ = that a polymonial + ∶ → can be added K− V V Relations: canbe decompased −+ ∶ → finitely many 0 V Properties: x, y A xy xx x y xx times tog et the into liner factors Relations: × → Zero Algebra additive identity Properties: ∈is associative and commutative ∀ ∈ ∶ ( )( ) = ( ( ( )) Characteristic Nonzero Field K V is scalar multiplication by K Types: An algebra A Types: A field K +V, is a group Algebra Over a Field Functions: Relations: Functions: Splitting Field × Types: A set of elements A,withunderlyingfieldK ( +) 2 Properties: u, v A, uv 0 Relations: Types: A field K , along with field K, polynomial ring K X Functions: A A the product of all Properties: n 1 1 ... 1 0 2 A is inherently associative and commutative N Functions: ′ A A elements in A is 0 ⋅∶ → ∀ ∈ = n Relations: [ ] K A A ∃ ∈ ∶ + + + = + ∶ → Properties: K is a algebraic extension of K Relations: ￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿ × → K′ of some x K X is the smallest extension so that x can be decomposed into linear factors Properties: is associative and commutative ′ is not assumed commutative or associative +distributes over is associative equipped with a ∈ [ ] A map from a K⋅ A represents scalar multiplication partial order, g to a unital ⋅ A, ,K is a vector+ space ⋅ All polynomials assocative algebra ≤ × can be decompased need not be ( + ) of a particular structure and a new mathematical structure. into linear factors commutative ⋅ ?? Types: An associative algebra and locally finite poset, A Finite Field Algebraically Closed Field Noncommutative Algebra Associative Algebra Functions: A A Types: A characteristic nonzero field K Types: A field K and a polynomial ring K X A made up of Types: An algebra A Types: An algebra A A2 A Functions: Functions: functions that Functions: Functions: Relations: ∶ → Relations: Relations: [ ] assign subinte- Relations: Relations: Properties: ∗∶is algebra→ multiplication Properties: K has a finite number of elements Properties: All polynomials x K X can be decomposed as the product of linear factors vals to scalars Properties: in A is need not be commutative Properties: is associative Any a A behaves such that b, c a b, c The order of some k K is pn for some p prime, n Equivalently, there is no proper algebraic extension of K A has an alphabet N The idenitity a∗ b, c is taken from a commutative ring of scalars ∈ [ ] Free Algebra of indeterminates ⋅ for is in A ⋅ is the∈ identity function [ ] ￿ ( ) ∈ ∈ Types: A noncommutative algebra A X ( ) Functions: 1 A X The symmetric ⋅ Relations: [ ] algebra is a free Properties: X∈ [x1,x] 2,...,xn ,xi is an indeterminate unital commu- 1 is the empty word tative algebra = { } Unital Algebra Members of A are Types: An algebra A linear opertors Functions: 1 A T V is the free Relations: Operator Algebra A equipped algebra over a Properties: 1∈ is the identity for Types: An associative Algebra A with a norm, (vector) space Functions: A2 A is the ten- ⋅ Relations: ￿￿ sor product Properties: a○∶ A are→ continuous linear operators over a topological vector space ⋅ is operator composition (and algebra multiplication) Tensor Algebra ∈ Banach Algebra ○ Types: A noncommutative Algebra T Types: An associative algebra A Functions: T 2 T Functions: A R T 2 T Relations: ￿￿ ￿￿ ∶ → The mathematical literature as well as relationship with other Relations: ⊕∶ → Properties: x, y A xy x y Properties: T⊗∶is defined→ over any vector space V A is also a Banach space k ∀ ∈ ∶ ￿￿ ￿￿ ≤ ￿￿ ￿￿ ￿￿ ￿￿ T V V 1 V 2 ... k V T V C V V V V V V .... T kV = T l⊗V T⊗k lV ⊗ T (V ) =is the⊕ most⊕ +( general⊗ ) ⊕ algebra( ⊗ over⊗ )V⊕ ⊗ = ( )

is a quoient is a quoient algebra of T V algebra of T V ￿ ￿ ( ) ( )

Exterior Algebra Universal Enveloping Algebra is a quoient Types: A nonassociative algebra V of a vector space V Types: A unital associative Algebra g of a Lie Group g algebra of T V Functions: Functions: ￿ Relations: ￿( ) Relations: U( ) ( ) Properties: is the ideal generated from w v v w, w,v V Properties: is the ideal generated from x y y x x, y x, y T g V T V g g I ⊗ + ⊗ ∀ ∈ I ⊗ − ⊗ − [ ]∀ ∈ ( ) Division by the anticommutator makes V anticommutative ￿( ) = ( )￿I U( ) = ⊗ )￿I g ’s un- ￿( ) derlying Lie AlgebraU( ) is abelian

Symmetric Algebra Types: An associative, commutative algebra V of a vector space V mathematical structure served as a guide for this distinction. Functions: 1 V Relations: ￿( ) Properties: ∈is￿ the( ) ideal generated from w v v w, w,v V V T V DivisionI by the commutator makes⊗ − V⊗ commutative∀ ∈ ￿( ) = ( )￿I ( ) For example, General Linear Groups are not included while ￿ Cyclic Groups are. Fig. 1. Map of the largest mathematical structures. The hierarchical structure of each map was further enhanced as follows.

Magma Types: AsetM Size of Nodes 2 Functions: M M M is cancellitive is associative Relations: Properties: ⋅∶u, v →M u v M ⋅ Nodes that were ‘higher’ on the hierarchy, meaning ∀ ∈ ∶ ⋅ ∈ Quasigroup Semigroup Types: A magma Q Types: A magma S of elements Functions: Q2 Q they were more general and less specific (e.g. set), Functions: Q2 Q Relations: 1￿R∶, 1L →Q Properties: is associative Relations: ￿∶ → Properties: Q is a∈ cancellative magma ⋅ were given larger area while more specific and struc- Q has an iden- Q has an iden- tity element, 1 tity element, 1 tured nodes (e.g. primitive group) were given smaller Monoid Loop Types: A semigroup S Types: A quasigroup Q Functions: 1 S Functions: 1 Q Relations: Relations: Properties: 1∈ is the identity for Properties: 1∈ is the identity for dimensions. 1 u u 1 u 1 x x 1 x ⋅ ⋅ ⋅ = ⋅ = ⋅ = ⋅ = Coloring

all elements in S have an is associative Different sections of structures were colored dif- inverse in S ⋅ ferently. Ten large sections were identified (alge-

Group bras, fields, graphs, groups, lattices, posets, modules, Types: AsetG Functions: G2 G 1 G G 1⋅∶− G → Relations: − ∶ → rings, sets, topological spaces). Each was given a Properties: is∈ associative 1 is the identity for ⋅ 1 is the inverse map for − ⋅ distinct color. − ⋅ Some structures, such as a topological group, should be clas-

G has a fi- nite number sified in more than one section: groups and topological spaces. of generators To deal with the coloring issue, nodes were simply shaded Finitely Generated Group Finitely Presented Group Types: A group G Types: A finitely generated group G Functions: G has fi- Functions: multiple colors through fading. In the example of topological Relations: nite relators Relations: Properties: G is of the form S R , S a generating set, R relators Properties: R n, n N S n, n N ￿ ￿ ￿ ￿ ￿ = ∈ group, the topological group node faded from orange (the color ￿ ￿ = ∈ for groups) to purple (the color for topological spaces). Finally, G has one all nodes on all maps were hyperlinked to their wikipedia generator pages, so a user may further explore mathematical concepts Cyclic Group Polycyclic Group Types: A group G with element a G has a chain of Types: A group P with subgroups Hi Cyclically Ordered Group Functions: cyclic subgroup Functions: Types: A group C Add a rela- Relations: whose quotients Relations: disscussed relatively informally. Functions: tion, ,, Properties: G can be generated from a and a set of relations R are also cyclic Properties: P is necessarily finitely presented Relations: ,, a, b, c C C C a, b, c G is necessarily commutative Hi i 1,2,..,n 1 Hi Hi 1 is cyclic [ ] Properties: a, b, c is cyclic, asymmetric, ∈{ − } + [ ] = {( ) ∈ × × ￿[ ]} { } ∶ ￿ transitive and total All finitely gen- [ ] G is a subgroup in erated subgroups another group V of G are cyclic n 2 V. RESULTS ≤ Metacyclic Group Virtually Cyclic Group Locally Cyclic Group Types: A group G Types: A group V with subgroup H Types: A group L Functions: Functions: Functions: Relations: Relations: Relations: Properties: n 2 Properties: H V H is cyclic Properties: H L,ifH is finitely generated, it is cyclic This project created eleven maps, totaling 187 unique V H n, n N ≤ ∃ ≤ ∶ ∀ ≤ structures. The map shown in Figure 1 contained only all ￿ ∶ ￿ = ∈ of the structures. The overlay labels the large catagories of structures that were included. Nine maps contained all of Fig. 2. An outline of connected structures. the related structures grouped under single color (with the exception of posets and lattices, which were combined into one map with two colors). The final map contained only the are sized differently based on hierarchy, and arranged strate- largest structures, meaning the structures that recieved their gically to encode information and all nodes are purposefully own color. colored orange, the color designated to groups. 2 depicts the structure of a simplified map of groups. It is This project was designed to simplify and accelerate the noted that all nodes have information about the structures they understanding of how abstract structures, stripped of their represent, nodes are connected as minimally as possible, nodes commonly placed disciplines, interact with and relate to each [9] M Hegarty, M. Kozhevnikov “Types of visual-spatial representations and Topological Group mathematical problem solving” Journal of Educational Psychology, vol. Types: A group G with topology τ 91 no. 4 2 [10] D Kirsh. Thinking with external representations, AI & Society, vol. 25, Functions: G G no. 4, 2010 1 G G [11] A. Arcavi “The role of visual representations in the learning of mathe- matics” Educational Studies in Mathematics, 2003 ⋅ ∶− → 1 G [12] C. Yuan. “Circle Yuan?s Answer to ‘Has Anyone Made a Con- Relations: − ∶ → cept Map or a Mindmap of All Known Mathematical knowledge???. ∈ 1 Quora. September 28, 2017. https://www.quora.com/Has-anyone-made- Properties: , are continuous mappings a-concept-map-or-a-mindmap-of-all-known-mathematical-knowledge. − [13] K. Voelkel. “Mindmap On Complex Analysis In One Variable?. Novem- Fig. 3. An individual node.⋅ − It has types of group and topology, so t is shaded ber 14, 2011. http://blog.konradvoelkel.de/2011/11/complex-analysis- both orange, in accordance with the group color scheme, and purple, to match mindmap/. topological space coloring. [14] E. Weisstein and I. Ford, “The Semantic Representation of Pure Mathematics.? Wolfram Blog, 22 Dec. 2016, blog.wolfram.com/2016/12/22/the-semantic-representation-of-pure- mathematics/. other. The use of labeled arrows between structures is meant to [15] J. Kepner and H. Jananthan, “Mathematics of big data: spreadsheets, clearly depict the relationship they had with each other, and the databases, matrices, and graphs, MIT Press, 2018 [16] H. Jananthan and J. Kepner, “Foundations of Mathematics”, in prepara- use of consistent definitions is meant to make the distinctions tion between the structures as clear as possible. Aesthetic features, [17] Wolfram Mathworld: The Web’s Most Extensive Mathematics Resource. such as coloring, were used to highlight important subtleties Wolfram Research, Inc http://mathworld.wolfram.com/ [18] Wikipedia: The Free Encyclopedia. Wikimedia Foundation Inc. Ency- that arise from certain structures. Furthermore, a hierarchical clopedia on-line. https://en.wikipedia.org/ structure was also created in these maps, which gives indi- [19] D. S. Dummit and R. M. Foote. Abstract Algebra. Third Edition. John viduals a framework to organize definitions of related but Wiley and Sons, Inc. 2004. [20] Cmap. Florida Institute for Human and Machine Cognition. separated structures in their mind through the use of differently cmap.ihmc.us sized nodes and inclusion of structures in the Types of other [21] n.d. “PGF And TikZ – Graphic Systems For TeX?. SourceForge. structures. Updated June 21 2018. https://sourceforge.net/projects/pgf/.

VI.SUMMARY The project created a unique representation of common mathematical objects aimed at young mathematical students and professionals in related fields with none to minimal background in pure mathematics. Eleven maps were created in total, with careful attention paid to the visual details of the map, as conveying a visual representation of abstract concepts was especially important in a field that lacks ample visual representations. A single map with all 187 nodes is posted online and available for use [1].

ACKNOWLEDGMENTS The authors wish to acknowledge the following individuals for their contributions and support: Alan Edelman, Vijay Gadepally, Chris Hill, and Lauren Milechin.

REFERENCES [1] www.mit.edu/%7Ekepner/GravelMathMap.pdf [2] W.H. Levie and R. Lentz, ”Effects of text illustrations: a review of research.” ECTJ (1982) 30: 195 [3] B. Rasken and K. Rolka. ”A picture is worth 1000 words- the role of visualization in mathematics learning”. University of Duisburg-Essen, Germany, 2006 [4] R. P. Flores. “Concept mapping in mathematics: tools for the develop- ment of cognitive and non-cognitive elements”. Conference on Concept Mapping. 2008. [5] Whiteley, Walter. “Visualization in Mathematics: claims and questions towards a research program”. 2004 [6] A. Bishop “Spatial abilities and mathematical education: a review” Educational Studies in Mathematics, vol. 11, no. 3, pp. 257-269, August 1980 [7] D. Tall. “Thinking through three worlds of mathematics”. Group for Psychology of Mathematics Education, vol. 4, pp 281-288, 2004 [8] R. Siegler. “Implications of the cognitive science research for mathe- matics education” Carnegie Mellon University, 2003